Periodicity in generations of automata

A class of automata which build other automata is defined. These automata are called Turing machine automata because each one contains a Turing machine which acts as its computer-brain and which completely determines what its offspring, if any, will be. We show that for the descendants of an arbitrary progenitor Turing machine automaton there are exactly three possibilities: (1) there is a sterile descendant after an arbitrary number of generations, (2) after a delay of an arbitrary number of generations, the descendants repeat in generations with an arbitrary period, or (3) the descendants are aperiodic. We also show what sort of computing ability may be realized by the descendants in each of the possibilities. Furthermore, it is determined whether there are effective procedures for distinguishing between the various possibilities, and the exact degree of unsolvability is computed for those decision problems for which there is no effective procedure. Lastly, we discuss the relevance of the results to biology and pose several questions.